Haven’t had time to post new formulae due to this lab and paper.

I figure this paper has enough well-documented formulae to make up for it. After all, it’s what I was doing rather than updating here.

This is the Beer-Lambert law, which models acoustic (and other radiative forms of) attenuation. In it, the amplitude, or intensity of the energy incident on an attenuator is shown as A0. A is the output intensity, while l is the thickness of the attenuator, and alpha is its attenuation coefficient in m^-1.

Energy can take many different forms. In these two cases, we’re looking at the thermal and individual photon energies. Both of these are very useful in calculating the amount of energy in an optical system, one in the case of a heat-dispersing system, the other in the case of any source of photons. E is, of course, the energy. kB is the Boltzmann constant, 1.3806504*10^-23 J/K, and T is the temperature in K. h is the Planck constant, 6.626*10^-34 J*sec, and nu is the frequency of the photon.

The Q factor is a measure of how damped an oscillator is. Q is dimensionless, and is expressed as a reduction of the ratio of stored energy to lost energy. This can be seen in this equation as the resonant frequency, fr, over the change in power, Delta P, multiplied by 2 pi, the reducing factor, and the stored energy, E. Right now, I’m working with microwaves in my Advanced Physics lab, so this is rather important. More on that can be found on my lab blog: http://peteinthelab.wordpress.com/

Mackenzie (10.1121/1.386919) has done an excellent job in showing what the depth, salinity, and temperature dependence of the speed of sound in seawater is in an empirical setting. In this case, vc is the speed of sound in the medium, T is temperature in °C, S is salinity in ppt, and z is depth in meters with the origin at the sea surface (depth is negative). In the above plot, the x axis is temperature, the y axis is depth, and the z axis is speed of sound. This plot assumes an average salinity, which is usually about 35 ppt.

The simplified version of the Fourier conjecture: s(t) is a signal that varies in the time domain, and the right side of the equation is its decomposition into a Fourier series, or an infinite series of sinusoids. A is the amplitude, omega (which looks like a cursive w) is the reduced frequency (f/2π), and phi is the phase. Each of the values changes in the series such that you wind up adding up a series of very different sinusoids. A great example is a square wave, which can be decomposed into odd harmonics of a fundamental. This means that a square wave has an A=0 for n is even, and omega2=2*omega1, etc. This is useful for everything from digital audio and image processing to RF transmission theory to population dynamics and mechanical systems.

Expect to see a new, hopefully useful, formula posted here every week this year. This is in keeping with one of my goals on 101in365 to “post another cool formula to my tumblr each week.”

These two equations show the resistance and resistor power rating for an LED current limiting resistor. This is true where Vcc is the source voltage, VLED is the threshold voltage, or forward voltage drop across the LED, ILED is the current through the LED (usually around 10-20 mA for most small LEDs), Rlimit is the resistance, and Plimit is the dissipated power. Remember that all of these assume a DC power supply, and are for simple resistance, though you can use the positive part of an AC waveform (greater than 0 V or ground) and complex impedance to sub in for Vcc and Rlimit respectively. Now go make some flashlights!